Multiscale Geometric Modeling of Ambiguous Shapes with : oleranced Balls and Compoundly Weighted alphashapes.
ABSTRACT Dealing with ambiguous data is a challenge in Science in general and geometry processing in particular. One route of choice to extract information from such data consists of replacing the ambiguous input by a continuum, typically a oneparameter family, so as to mine stable geometric and topological features within this family. This work follows this spirit and introduces a novel framework to handle 3D ambiguous geometric data which are naturally modeled by balls. First, we introduce {\em toleranced balls} to model ambiguous geometric objects. A toleranced ball consists of two concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries. We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned radius interpolation is tantamount to the growth process associated with an additivelymultiplicatively weighted Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the CW diagram and the associated CW $\alpha$complex, which provides a filtration called the $\lambda$complex. Fourth, we propose a naive algorithm to compute the CW VD. Finally, we use the $\lambda$complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities.

Article: The geometric stability of Voronoi diagrams in normed spaces which are not uniformly convex
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ABSTRACT: The Voronoi diagram is a geometric object which is widely used in many areas. Recently it has been shown that under mild conditions Voronoi diagrams have a certain continuity property: small perturbations of the sites yield small perturbations in the shapes of the corresponding Voronoi cells. However, this result is based on the assumption that the ambient normed space is uniformly convex. Unfortunately, simple counterexamples show that if uniform convexity is removed, then instability can occur. Since Voronoi diagrams in normed spaces which are not uniformly convex do appear in theory and practice, e.g., in the plane with the Manhattan (ell_1) distance, it is natural to ask whether the stability property can be generalized to them, perhaps under additional assumptions. This paper shows that this is indeed the case assuming the unit sphere of the space has a certain (nonexotic) structure and the sites satisfy a certain "general position" condition related to it. The condition on the unit sphere is that it can be decomposed into at most one "rotund part" and at most finitely many nondegenerate convex parts. Along the way certain topological properties of Votonoi cells (e.g., that the induced bisectors are not "fat") are proved.12/2012;
Page 1
apport ?
?
de recherche?
ISSN 02496399
ISRN INRIA/RR7306FR+ENG
Thème BIO
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Multiscale Geometric Modeling of Ambiguous
Shapes with Toleranced Balls and Compoundly
Weighted αshapes
Frédéric Cazals — Tom Dreyfus
N° 7306
May 2010
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Unité de recherche INRIA Sophia Antipolis
2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
Multiscale Geometric Modeling of Ambiguous Shapes with
Toleranced Balls and Compoundly Weighted αshapes
Fr´ ed´ eric Cazals∗, Tom Dreyfus†
Th` eme BIO — Syst` emes biologiques
Projet ABS
Rapport de recherche n° 7306 — May 2010 — 29 pages
Abstract:
particular. One route of choice to extract information from such data consists of replacing the ambiguous input
by a continuum, typically a oneparameter family, so as to mine stable geometric and topological features within
this family. This work follows this spirit and introduces a novel framework to handle 3D ambiguous geometric
data which are naturally modeled by balls.
First, we introduce toleranced balls to model ambiguous geometric objects. A toleranced ball consists of two
concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries.
We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned
radius interpolation is tantamount to the growth process associated with an additivelymultiplicatively weighted
Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the
CW diagram and the associated CW αcomplex, which provides a filtration called the λcomplex. Fourth, we
propose a naive algorithm to compute the CW VD. Finally, we use the λcomplex to assess the quality of models
of large protein assemblies, as these models inherently feature ambiguities.
Dealing with ambiguous data is a challenge in Science in general and geometry processing in
Keywords:
molecular complexes, interfaces, structural biology.
Union of balls, Voronoi diagrams, αshapes, stability, topological persistence, proteins, macro
∗INRIA SophiaAntipolisM´ editerran´ ee, AlgorithmsBiologyStructure; Frederic.Cazals@sophia.inria.fr
†INRIA SophiaAntipolisM´ editerran´ ee, AlgorithmsBiologyStructure; Sebastien.Loriot@sophia.inria.fr
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Mod´ elisation Multiechelle de Formes Ambigu¨ es avec des Boules
Tol´ eranc´ ees et les αshapes ` a Pond´ eration Compos´ ee
R´ esum´ e :
exacerb´ e en g´ eom´ etrie. Une fa¸ con int´ eressante d’extraire de l’information de telles donn´ ees consiste ` a rem
placer cellesci par un continuum, typiquement une famille ` a un param` etre, de fa¸ con ` a chercher des structures
g´ eom´ etriques et topologiques stables au sein de cette famille. Ce travail s’inscrit dans cette veine, et propose
un nouveau canevas pour manipuler des donn´ ees 3D ambigu¨ es.
Tout d’abord, nous introduisons les boules toleranc´ ees. Un telle boule est constitu´ ee de deux boules con
centriques, et interpoler entre leurs rayons permet d’explorer un ensemble de g´ eom´ etries possibles. Nous pro
posons de mod´ eliser une forme ambigu¨ e par un ensemble de boules toleranc´ ees, et montrons que le proces
sus d’interpolation ´ evoqu´ e cidessus conduit ` a un diagramme de Voronoi additifmultiplicatif (aussi appel´ e ` a
pond´ eration compos´ ee, ou CW). Ensuite, nous nous int´ eressons aux propri´ et´ es du diagramme CW et ` a l’αshape
associ´ ee, qui repr´ esente une filtration que nous nommons le λcomplexe. Nous poursuivons par un algorithme
na¨ ıf de calcul du diagramme de Voronoi CW. Enfin, nous montrons comment utiliser le λcomplexe pour ´ etudier
la qualit´ e de mod` eles de gros assemblages macromol´ eculaires, ces mod` eles pr´ esentant de fa¸ con intrins` eque des
ambigu¨ ıt´ es.
La manipulation de donn´ ees ambigu¨ es est un challenge tout ` a fait g´ en´ eral, qui est particuli` erement
Motscl´ es :
complexes macromol´ eculaires, interfaces, biologie structurale.
Union de boules, diagrammes de Voronoi, αshapes, stabilit´ e, persistence topologique, prot´ eines,
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Modeling with toleranced balls3
1Introduction
1.1 Voronoi Diagrams and Applications
The Voronoi diagram of a finite collection of sites equipped with a generalized distance is the cell decomposition
of the ambient space into equivalence classes of points having the same nearest sites for this distance. Voronoi
diagrams are central constructions in science and engineering [OC00], and their versatility actually comes from
two sources. First, the great variability of sites and distances is a first source of diversity. While the most
classical construction is the Euclidean distance diagram for points, the mere class of circles and weighted points
yields as diverse diagrams as power, Apollonius and M¨ obius diagrams–see [BWY06] and Fig. 1. Second, the
information encoded in a Voronoi diagram actually goes beyond the aforementioned cell decomposition. One
can indeed consider the bisectors bounding the cells as the realization of a growth process defined by the
distance, in the sense that the level sets of this distance intersect on the bisectors. This viewpoint motivated
the development of αcomplexes and αshapes [Ede92], a beautiful construction providing a filtration of the
Delaunay triangulation dual of the Voronoi diagram, later complemented by the flow complex [GJ03]. From
a mathematical standpoint, these developments are concerned with the topological changes undergone by the
sublevel sets of the distance, which is the heart of Morse theory [Mil63]. Ideas in this realm also motivated
the development of topological persistence [ELZ02, CSEH05], a subject concerned with the assessment of the
stability of topological features associated with the sublevel sets of a function defined on a topological space.
The success of αshapes relies on two cornerstones. First, the aforementioned growth process gives access to
a multiscale analysis of the input sites. For example, the problem of reconstructing a shape from sample points
can be tackled by considering the spacefilling diagram consisting of balls grown around the sample points. Alas,
a provably good reconstruction using this strategy requires a uniform sampling, which motivated the definition
of more local growth processes. One may cite conformal αshapes [CGPZ06], where the growth process depends
on the distance of the samples to the medial axis, and the scaleaxis transform [GMPW09] which provides a
hierarchy of skeletal shapes based on a dilation (and retraction) of medial balls. Second, αshapes inherently
model objects represented by collections of balls—in particular molecules, and encode remarkable geometric
and topological properties [Ede95]. For the particular case of proteins and small macromolecular complexes,
they have been instrumental to compute molecular surfaces [AE96] and model interfaces [BGNC09, LC10].
Our developments are precisely motivated by the requirement to perform multiscale analysis in computa
tional structural biology.
1.2Contributions
Structural proteomics studies are concerned by the identification and the modeling of molecular machines
operating within the cell [GAG+06], and a major endeavour consists of modeling assemblies involving from tens
[LTSW09] to hundreds [ADV+07] of polypeptide chains. This modeling requires integrating data coming from
several experimental sources, these data being typically noisy and incomplete [AFK+08]. In this context, the
premises just discussed on αshapes certainly hold: on the one hand, balls are the primitive of choice since we
deal with atoms and molecules; on the other hand, multiscale analysis are in order since we deal with ambiguous
data and need to accommodate uncertainties on the shapes and positions of proteins within an assembly. In
this context, we make the following contributions.
First, we introduce toleranced balls to model ambiguous geometric shapes. A toleranced ball consists of two
concentric balls—the inner and outer balls, and interpolating between them allows one to replace an arbitrary
ball by oneparameter family of balls. As illustrated on Fig. 2, the inner (outer) ball of a toleranced ball
is meant to accommodate high (low) confidence regions. We note in passing that our approach bears some
similarities with modeling with toleranced parts in engineering, where tolerances are generally accommodated
thanks to Minkowski sums [LWC97]. Second, we show that the growth process associated with this interpolation
is associated with a socalled additivelymultiplicativelyweighted Voronoi diagram, also called compoundly
weighted Voronoi diagram—CW VD for short. Third, we investigate properties of this CW diagram and its
dual. Fourth, we present the filtration, called the λcomplex, induced on the dual by the growth process, and
encoding all possible topologies associated with this growth process. For any value of λ, the λcomplex identifies
a list of simplices of the dual complex, together with a label for each of them. This label precisely encodes
the contribution of the simplex to the boundary of the union of the balls grown thanks to the additively
multiplicatively model. Our analysis generalizes the socalled βshapes [SCC+06], i.e. the αshape associated
to an Apollonius diagram [BD05, BWY06, EK06]. Fifth, we present an algorithm to compute the CW VD.
Finally, we present experimental results on toleranced protein models.
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Figure 1: Curved Voronoi diagrams of 7 circles / weighted points. (Left) Power diagram d(Si(ci,ri),p) =
?p − ci?2− r2
d(Si(ci,ai,ri),p) = ai?p − ci?2− r2
i. (Middle) Apollonius diagram : d(Si(ci,ri),p) = ?p − ci? − ri (Right) M¨ obius diagram :
i.
Figure 2: A fictitious molecule of three atoms undergoing conformational changes.(a) The two extreme confor
mations, together with the probability density map D in background. The map D displays the probability for
a given point to be covered by a random conformation. (b) Three toleranced balls used to cover the portion of
the map D involving probabilities beyond a given threshold. Dashed lines represent the inner and outer balls
of the toleranced balls. Note that higher the confidence / probability, the smaller the region between the inner
and outer balls.
2Toleranced Models and Compoundly Weighted Voronoi Diagram
2.1Compoundly Weighted Distance and Toleranced Balls
Toleranced balls.
and αi, we define the additivelymultiplicatively distance as follows:
Given a weighted point Si(ci;µi,αi), with center ciand parameters (real numbers) µi> 0
λ(Si,p) = µi cip  −αi.
(1)
This distance is associated with socalled compoundlyweighted Voronoi diagrams or CW VD for short [OC00].
Geometrically speaking, this distance is best understood using the following growth process. Let a toleranced
ball Si(ci;r−
and outer balls. Given a toleranced ball Siand a real parameter λ, consider the grown ball Si[λ] centered at ci
and whose radius is defined by:
ri(λ) = r−
i,r+
i) be a pair of concentric balls of radii r−
i< r+
i, centered at ci. These balls are called the inner
i+ λ(r+
i− r−
i).
(2)
Denoting δi= r+
i− r−
i, a point p is reached by this growth process once ri(λ) = cip , that is
λ(Si,p) = cip 
δi
−r−
i
δi
.
(3)
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Modeling with toleranced balls5
In other words, a toleranced ball Si(ci;r−
and reciprocally, a weighted point Si(ci;µi,αi) is tantamount to a toleranced ball Si(ci;r−
(1 + αi)/µi). In the sequel, we shall use both terminologies and exchangeable refer to weighted point Si or
toleranced ball Si.
i,r+
i) is tantamount to a weighted point Si(ci;µi= 1/δi,αi= r−
i/δi);
i
= αi/µi,r+
i
=
2.2On Concomitant Interpolation Processes
Consider two toleranced balls Si and Sj. We term the linear interpolation of Eq. (2) concomitant since at
λ = 0 (resp. λ = 1) the grown balls Si[λ] and Sj[λ] respectively match their inner (outer) balls. In the
context of toleranced models, concomitance is important since, for a collection of toleranced balls, we aim
at exploring the region sandwiched between the inner and outer balls coherently. Interestingly, concomitance
requires multiplicatively weighted Voronoi diagram — CW or M¨ obius.
Non concomitant interpolations.
squared radius as follows:
For the power diagram, the growth process consists of modifying the
r2
i(α) = cip 2= r2
i+ α.
(4)
Let a toleranced weighted point be a pair of concentric balls of weights r2
αi required to interpolate from the inner to the outer ball is αi = (r+
concomitant since for two toleranced weighted points, one generically has αi?= αj.
i= (r−
i)2. The interpolation is not
i)2and (r+
i)2. The value
i)2− (r−
The same observation holds for the growth process associated with an Apollonius diagram, which is not
concomitant unless the discrepancy r+
i− r−
iof all toleranced balls is equal to some constant.
Concomitant interpolations.
diagrams, recall that the generalized M¨ obius distance to a weighted point Si(ci,µi,αi) is defined by:
To see that M¨ obius diagrams share the concomitance property with CW
d(Si,p) = µi cip 2+αi.
(5)
Equivalently,
 cip 2=
1
µi(d + αi).
(6)
To make the connexion between the distance of Eq. (5) and a toleranced ball, we use d = 0 and d = 1, which
yields
(r−
µi, and (r+
Equivalently, one has:
1
(r+
i)2=αi
i)2=1 + αi
µi
.
(7)
µi=
i)2− (r−
i)2and αi=
(r−
i)2
(r+
i)2− (r−
i)2.
(8)
A comparison of the CW and M¨ obius growth models, that is ri(λ) = cip  versus ri(d) =
provided on Fig. 3. Compared to the CW linear growth model and as shown by the variation of the derivative
of ∂ri(d)/∂d, a large difference (r+
? cip 2, is
i)2− (r−
i)2biases the M¨ obius interpolation towards small values.
2.3Toleranced Tangency and Generalization of the Empty Ball Property
For affine (Apollonius) Voronoi diagrams, it is well known that for each point centered on a Voronoi face, there
exists a unique ball orthogonal (tangent) to the balls associated with the vertices of the dual simplex, and
conflict free with all the other balls1. To derive the analogue in the CWcase, consider a point p and two
toleranced balls Siand Sjsuch that λ(Si,p) = λ < λ(Sj,p). For the pair Siand p, one gets with Eq. (1):
 pci −αi
µi
−λ
µi
= 0 ⇔  pci −r−
i− λδi= 0.
(9)
1Consider e.g. the power case, and pick a point p on the Voronoi face dual of a simplex involving a ball Si(ci,wi). Assume that
point p lies on the sphere bounding the ball Si(ci,wi+ α). One has π(p,Si) = cip 2−wi− α = 0, or equivalently, the balls Si
and X(p,α) are orthogonal.
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Figure 3: Comparing the variation of the radius for the CW model (green curve) and the M¨ obius model (red
curve). On this example, r−
i= 0 and r+
i= 10.
Similarly, for the pair Sjand point p:
 pcj −αj
µj
−λ
µj
> 0 ⇔  pcj −r−
j− λδj> 0.
(10)
We summarize with the following definition, illustrated on Fig.4:
Definition. 1. A ball B(p,λ) which satisfies the condition of Eq. (9) w.r.t. a toleranced ball Si is called
toleranced tangent (TT for short) to Si. A toleranced ball Sj and a ball B(p,λ) which satisfy the condition of
Eq. (10) are called conflict free.
Remark. 1. Equation (9) states that the inner ball B(ci,r−
scaled by δi, are tangent. Similarly, condition (10) states that B(cj,r−
shall use this property to illustrate TT balls, see e.g. Fig. 4.
i) and the ball B(p,λδi)—which is the ball B(p,λ)
j) and B(p,λδj) do not intersect. We
Remark. 2. Let S be a collection of toleranced balls. Consider a ball B(p,λ) which is TT to a subset of balls
T ⊂ S, and conflict free with the toleranced balls in S\T. The center p of this ball is found at the intersection
of the spheres bounding the grown balls Si[λ] with Si ∈ T, and is located outside the grown balls Sj[λ] with
Sj∈ S\T.
S1
S2
S3
p
Figure 4:
S1(0,0;1,5),S2(0,10;2,8),S3(4,−9;1,3). The three dotted circles represent S1[3/4],S2[3/4],S3[3/4]. The three
circles centered at p are the scaled versions of ball B(p,3/4); following remark 1, ball B(p,3/4) is TT to S1and
S2, and conflict free with S3.
Toleranced tangent (TT) balls and conflict free balls. In dashed lines, toleranced balls
Remark. 3. In Eq. (9) and (10), the radius of the toleranced ball B(p,λδi) depends on the parameter δifrom
toleranced ball Si. Denoting δ+the additively weighted distance between two weighted points, Eq. (9) and Eq.
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Modeling with toleranced balls7
(10) may be rewritten as follow:
δ+(B(p,λ),B(pi,r−
i
δi
)) =δi− 1
δi
 ppi,
(11)
and
δ+(B(p,λ),B(pi,r−
i
δi
)) >δi− 1
δi
 ppi .
(12)
The lefthand side involves B(p,λ), a ball whose radius does not depend on parameters from toleranced balls of
S, as for the power and Apollonius cases. But the righthandside is parametrized. In the sequel, we use Eq.
(9) and Eq. (10) for a simpler geometric interpretation of toleranced tangency and conflictness. A generic ball
not belonging to S will be denoted B(p,λ).
3The Compoundly Weighted Voronoi Diagram
Consider a collection S of n toleranced balls. The Compoundly Weighted Voronoi diagram is the partition of
the space according to the nearest neighbor relationship, for the CW distance, that is:
V or(Si) = {p ∈ R3 λ(Si,p) ≤ λ(Sj,p)∀j ?= i}.
(13)
More generally, denoting Tk+1a tuple of k+1 toleranced balls, we are interested in V or(Tk+1) = ∩Si∈Tk+1V or(Si).
Naturally, we are also interested in the dual complex generalizing the Delaunay triangulation.
3.1Bisectors in the CW Case
The bisector of a tuple of toleranced balls Tk+1is the loci of points having the same CW distance w.r.t. every
toleranced ball. We denote this bisector ζ(Tk+1), and examine in turn the case for pairs, triples, and quadruples.
Our analysis assumes that the δiare not equal, as this is the Apollonius case [BWY06].
3.1.1Bisector of two toleranced balls
Analysis.
ζ(i,j) of Siand Sj:
Let Siand Sjbe two toleranced balls. The following property describes the existence of the bisector
Proposition. 1. Siis trivial w.r.t. toleranced ball Sjiff δi≤ δjand the following condition, which states that
cibelongs to the interior of the Voronoi region of Sj, holds:
λ(Sj,ci) < −r−
i
δi
.
(14)
Proof. If the Voronoi region Viof Siis empty, one has in particular, ci?∈ Vi, which is exactly Eq. (14). The
second implication also trivial holds. For the converse, applying the definition of λ(Si,p) to any point p, we get:
λ(Si,p) = pci −r−
i
δi
> pci
δi
+ cicj −r−
j
δj
(15)
≥ pci +  cicj −r−
> pcj −r−
δj
j
δj
(16)
j
= λ(Sj,p).
(17)
The three derivations respectively stem from Eq. (14), from δi≤ δj, and from the triangle inequality.
Assuming that ζ(i,j) exists, its geometry depends on the relative values of δi and δj. Assuming w.l.o.g.
that δi< δj, Sjgrows faster than Siso that for a large enough value of λ, the grown ball Si[λ] is contained in
its counterpart Sj[λ], so that the bisector is a closed surface, with ciin the bounded region delimited by ζ(i,j).
Matching the generalized distances shows that this surface is a degreefour algebraic surface. See Fig. 5 for a
2D illustration.
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Figure 5: Two toleranced balls and their bisector which is a degree four algebraic curve –green curve. Dashed
circles corresponding to the inner and outer balls. Dotted circles correspond to the solutions of a degree four
equation : blue ones are toleranced tangent circles, red ones are algebraic artifacts.
Extremal TT balls.
λ value is a local extremum. By radial symmetry w.r.t. the line joining the centers of the balls, such balls are
necessarily centered at the intersection between the bisector and the line joining the centers. Assume w.l.o.g.
that δi< δj. The minimal such ball, denoted Mi,j(mi,j,ρi,j) is such that Si[ρi,j] and Sj[ρi,j] are tangent at
mi,j. The maximal ball Mi,j(mi,j,ρi,j) is such that Si[ρi,j] is interiortangent to Sj[ρi,j] at mi,j.
If the bisector exists, it makes sense to track the TT balls such that the corresponding
Remark. 4. As illustrated on Fig. 6, Sj[ρi,j] may be exterior or interior to Si[ρi,j]; Ball Sj[ρi,j] is interior
to Si[ρi,j] iff Si is closer to cj than Sj for the CW distance i.e. λ(Si,cj) < λ(Sj,cj). In the limit case
λ(Si,cj) = λ(Sj,cj), cj= mi,jand Sj[ρi,j] may be considered as exterior to Si[ρi,j].
cj
ci
mi,j
mi,j
mi,j
mi,j
cj
ci
Figure 6: Relative position of mininal and maximal TT balls of two balls (Left) Sj[ρi,j] and Si[ρi,j] are exterior
tangent (Right) Sj[ρi,j] is interior tangent to Si[ρi,j].
The parameters of these extremal TT balls are computed as follows:
Proposition. 2. The two extremal TT balls B(p,λ) of two toleranced balls are characterized by
λ = cicj −(αr−
i+ βr−
j)
αδi+ βδj
,
(18)
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Modeling with toleranced balls9
and
? cip = αλδi+ r−
i
 cicj
?cicj,
(19)
where α = ±1 and β = ±1 depend on the ball processed (minimal or maximal) and the relative positions of Si
and Sj(case analysis in the proof).
Proof. Denote ? up,p? the unit vector between two points p and p?. The extremal TT ball Mi,j= (p,λ) or
Mi,j= (p,λ) of Siand Sjbeing centered on the line joining the centers ciand cj, we can express the weight λ
as follows:
? cip + ? pcj= ?cicj
(20)
⇔ cip  ? ucip+  pcj ? upcj= cicj ? ucicj
(21)
⇔ cip  ? ucip.? ucicj+  pcj ? upcj.? ucicj= cicj
(22)
α(λδi+ r−
i) + β(λδj+ r−
j) = cicj,
(23)
where α = ? ucip.? ucicj= ±1 and β = ? upcj.? ucicj= ±1. Equation (18) follows easily. We note in passing
that following remark 4, the signs of the dot products α and β are obtained from the sign of the expression
λ(Si,cj) − λ(Sj,cj).
The weight of the extremal TT balls being determined, the center is computed as follows:
α? ucip= ? ucicj
(24)
⇔ α
? cip
 cip =
⇔ ? cip = α cip 
?cicj
 cicj
(25)
 cicj
?cicj
(26)
⇔ ? cip = αλδi+ r−
i
 cicj
?cicj.
(27)
3.1.2Bisector of three toleranced balls
Analysis.
the Apollonius case, we suppose without loss of generality that δi0≤ δi1≤ δi2with δi0< δi2. If there is no
intersection between ζ(i0,i1) and ζ(i0,i2), ζ(i0,i1,i2) does not exist, and reciprocally. Assume that ζ(i0,i1,i2)
exists. Since at least one δidiffers from the other two, there is at most one Apollonius bisector. The geometry
of ζ(i0,i1,i2) depends on δi0,δi1and δi2, and the following cases are illustrated on Fig. 7.
? CWB.III.1 If there is no Apollonius bisector, ζ(i0,i1,i2) is a bounded curve resulting from the intersection
of two CW bisectors.
? CWB.III.2 If the Apollonius bisector is not a half straight line, ζ(i0,i1,i2) is a bounded curve resulting from
the intersection of one CW bisector, and one sheet of a hyperboloid (possibly degenerated to a hyperplane).
? CWB.III.3 If the Apollonius bisector is a half straight line, ζ(i0,i1,i2) is reduced to at most two intersection
points. Note that if there are two intersection points, δi1= δi2and Si1is included in and tangent to Si2.
Consider three toleranced balls Si0,Si1,Si2such that the bisector of each pair exists. To avoid
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10Cazals and Dreyfus
Figure 7: Bisectors of three toleranced balls. The red dots are the centers of the toleranced balls and the
pink/green/blue surfaces respectively represent the bisectors ζ(0,1) / ζ(0,2) / ζ(1,2). (Left) CWB.III.1
No Apollonius bisector (Middle) CWB.III.2 One Apollonius bisector (Right) CWB.III.3 One degenerate
Apollonius bisector.
Extremal TT balls.
half straight line, these balls are found by intersecting this line with one of the other two bisectors.
In the general case, identifying these two balls involves four equations in four unknowns—the coordinates of
the center and the weight λ. Denote π the plane defined by the centers of the three balls. The growth of the
balls being symmetric with respect to this plane, the fourth equation consists of constraining the center of an
extremal TT ball to plane π. The calculation is covered by the following proposition for k = 2:
In any case, there are two (possibly identical) extremal TT balls. If one bisector is a
Proposition. 3. Let Tk+1= {Sij}j=0,...,k be a triple or quadruple of toleranced balls, i.e. k = 2 or k = 3.
Computing the two extremal TT balls of the tuple Tk+1requires solving a degree four equation. A value solution
λ of this equation is valid provided that λδij+ r−
ij≥ 0, ∀j = 0,...,k.
Proof. of Prop. 3 for Tk+1= {Si0,Si1,Si2}.
The ball sought has to be TT to each of the three toleranced balls, as specified by Eq. (9). Assume that the
plane containing the centers of the three balls has equation axx+ayy+azz = aC. Squaring the three equations
of toleranced tangency yields the system:
Subtracting the first squared equation from the two subsequent ones yields:
Using Gaussian elimination on the last three equations, one obtains three linear equations for the coordinates of
p, parametrized by λ2. Injecting these quantities into the first equation yields the quartic equation in λ. Note
that a solution is valid iff λδij+ r−
that the coordinates of point p are rational fractions in λ2.
(p − ci0)2= (λδi0+ r−
(p − ci1)2= (λδi1+ r−
(p − ci2)2= (λδi2+ r−
axx + ayy + azz = aC
i0)2
i1)2
i2)2
(28)
(p − ci0)2= (λδi0+ r−
2p(ci1− ci0) = (λδi0+ r−
2p(ci2− ci0) = (λδi0+ r−
axx + ayy + azz = aC
i0)2
i0)2− (λδi1+ r−
i0)2− (λδi2+ r−
i1)2− (c2
i2)2− (c2
i0− c2
i0− c2
i1)
i2)
(29)
ij≥ 0, and sorting the valid values yields the extreme TT balls. Also note
Remark. 5. Geometrically, three intersecting spheres generically intersect into two points. The extreme TT
balls correspond to the situations where these two points coalesce.
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Modeling with toleranced balls11
3.1.3Bisector of four toleranced balls
Analysis.
the Apollonius case, we suppose w.l.o.g. that δi0≤ δi1≤ δi2≤ δi3with δi0< δi3. If the intersection of ζ(i0,i1),
ζ(i0,i2) and ζ(i0,i3) is empty, the intersection of all bisectors of pairs is empty and ζ(i0,i1,i2,i3) does not exist,
and reciprocally. If ζ(i0,i1,i2,i3) exists, we have ζ(i0,i1,i2,i3) = ζ(i0,i3)∩ζ(i1,i2,i3), from which the following
analysis follows.
? CWB.IV.1 The bisectors ζ(i0,i3) and ζ(i1,i2,i3) being a surface and a curve, their generic intersection, if
any, consists of a finite set of points. As we shall see below, there are at most four such points.
? CWB.IV.2 As a degenerate case, when ζ(i1,i2,i3) is a bounded curve, the intersection of ζ(i1,i2,i3) and
ζ(i0,i3) may be ζ(i1,i2,i3). In this case, ζ(i0,i1,i2,i3) has the geometry of the bisector ζ(i1,i2,i3) of three
toleranced balls.
Consider four toleranced balls Si0,Si1,Si2,Si3such that the bisector of each pair exists. To avoid
Extremal TT balls.
toleranced balls, we refer to the analysis carried out in section 3.1.2. Otherwise, ζ(i0,i1,i2,i3) is reduced to at
most four points, as shown by the following constructive proof of proposition 3:
We distinguish two cases. If ζ(i0,i1,i2,i3) has the geometry of a bisector of three
Proof. of Prop. 3 for Tk+1= {Si0,Si1,Si2,Si3}.
The ball sought has to be TT to each of the four toleranced balls, that is  pcij= λδij+ r−
Squaring the four equations of toleranced tangency yields the system
As for the case of three toleranced balls, we use Gaussian eliminations on system (30) to get three equations
linear to the coordinates of p and parametrized by λ2, and one quartic equation on λ. Checking that  pcij≥ 0
provides the valid solutions, and sorting these provides the extremal solutions.
ijfor j = 0,1,2,3.
(p − ci0)2= (λδi0+ r−
(p − ci1)2= (λδi1+ r−
(p − ci2)2= (λδi2+ r−
(p − ci3)2= (λδi3+ r−
i0)2
i1)2
i2)2
i3)2
(30)
Remark. 6. As illustrated on Fig. 8, the system (30) may have four distinct solutions λisuch that λiδj+r−
0.
i≥
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Before intersection: λ = λi− ?
After intersection: λ = λi+ ?
Figure 8: Upon growing, four toleranced balls may intersect into four distinct points. Denoting ? an arbitrarily
small number, we display the toleranced balls Si[λj± ?]. The λjhave been sorted by increasing value from Top
to Bottom.
3.2Voronoi Diagram and its Dual Complex
Empty Voronoi regions.
condition of triviality for two toleranced balls. But triviality of a toleranced ball amidst a collection of balls
is more complex since a toleranced ball might not be trivial w.r.t. any other one, yet, it might be trivial with
respect to their union. To see why, observe that Eq. (13) tells us that a point in space is attributed to the
Voronoi region of a toleranced ball provided that this toleranced ball reaches this point first in the growth
process. Thus, a growing ball which is always contained in the union of a collection of growing balls is trivial,
although it might not be trivial with any of them. Denoting T a collection of toleranced balls, and for any value
of λ, the following condition, illustrated on Fig. 10, must hold for Sito be trivial:
?
A toleranced ball whose region is empty is called trivial. Proposition 1 gives a
Si[λ] ⊂
Sj∈T
Sj[λ].
(31)
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Modeling with toleranced balls13
Remark. 7. The triviality condition is more complex than in the Apollonius case, where a ball is hidden if and
only if it is included within another ball.
Figure 9: Dual complex of the four balls of Fig.
8—bottom and top rows respectively represent 0
simplices and 3simplices.
c0
c1
c2
Figure 10:
(0,1/2;1,3) (red), S1 = (0,0;3,4) (green) and
S2 = (5,0;3,7) (blue). Ball S0 is neither trivial
w.r.t.
S1 nor S2, but is trivial with respect to
both.
Hidden toleranced ball.
S0
=
Dual Complex.
each being termed a face. Each such face corresponds to the intersection of k + 1 Voronoi regions, so that we
associate an abstract simplex or simplex for short in the dual complex. That is, if V or(Tk+1) consists of m faces,
on finds ∆j(Tk+1),j ∈ 1,...,m simplices in the dual complex. (The multiplicity is omitted if the tuple Tk+1
yields a single simplex.) The dual of a simplex ∆(T) is denoted ∆(T)∗. Assuming that the input toleranced
balls are numbered from 1 to n, a simplex is identified by a list of integers, and inclusion between such lists
defines a partial order on simplices. We therefore represent the dual complex by a Hasse diagram DSwith one
node per simplex. The nodes of DS corresponding to ksimplices are denoted DS(k). Note that we may also
(arbitrarily) embed a simplex within the union of Voronoi faces it is associated with. See Figs. 11, 12 and 13
for a 2D illustration.
The Voronoi region V or(Tk+1) of a tuple Tk+1 may have several connected components,
Topological complications.
ing toleranced ball defines a lens between the Voronoi region of two neighboring toleranced balls, a case also
found in the Apollonius diagram. In the dual complex, the vertex of this toleranced balls has exactly two
neighbors and the triangle corresponding to these three toleranced balls does not have any coface.
A Voronoi region might not be connected, and this may happen for tuples of size one to four. We illustrate
this in 2D with Fig. 11. For a toleranced ball, consider S4whose Voronoi region is split into two faces, associated
with the vertices (zerodimensional simplices) ∆1(4) and ∆2(4) in the Hasse diagram. For two toleranced balls,
note that the Voronoi region V or(S1,S2) consists of two faces—open line segments in this case, yielding the
simplices ∆1(1,2) and ∆2(1,2) in the Hasse diagram. For three toleranced balls, note that the triple (S1,S2,S4)
corresponds to two triangles.
A Voronoi region may not be simply connected. When one toleranced ball punches a hole into a face, the
corresponding onesimplex does not have any coface. See e.g. toleranced ball S7and the simplex ∆(2,7) on
Fig. 11. When two toleranced balls punch a hole into a Voronoi region, the twosimplex they define does not
have any coface either. Finally when three toleranced balls punch a hole into a Voronoi region, two tetrahedra
of the dual complex share the same vertices, the same edges and same triangles. This latter case is illustrated
in 2D, where a hole punched by two toleranced balls in a Voronoi region results in two triangles with the same
vertices and the same edges. See ∆1(2,5,6) and ∆2(2,5,6) on Fig. 11.
A Voronoi region gets sandwiched between two neighbors when the correspond
Bounded and Unbounded Voronoi regions.
δi≥ δj,∀j ?= i. A toleranced ball which is not maximal has a bounded Voronoi region in the CW VD of S, and
the subset of maximal toleranced balls is denoted Smax. The CW VD diagram of toleranced balls in Smaxis
an Apollonius diagram since all δiare equal, and a subset of balls in Smaxhave an unbounded Voronoi region.
A toleranced ball Si∈ S is called maximal w.r.t. to S if
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